Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimum-Weight Path in a Sparse Erdős--Rényi Graph with Signed Weights

Heng Ma, Pascal Maillard

TL;DR

This work extends minimal-weight path analysis on sparse Erdős–Rényi graphs to signed edge weights, under a tilt condition that ensures path weights grow with length. By combining renewal theory, local weak convergence to branching random walks, and Chen–Stein Poisson approximation, the authors prove that the rescaled extremal pairs (weight, hopcount) converge to a Cox process with random intensity given by the product of two independent Biggins BRW martingale limits. The results generalize previous nonnegative-weight findings, handling sign-induced complexities via a localized BRW framework and a careful conditioning scheme. The approach yields a precise extremal process description and accommodates lattice vs non-lattice weight distributions, providing a robust method for first-passage percolation on sparse random graphs with general weights.

Abstract

We consider a sparse Erdős--Rényi graph $\mathcal{G}(n,λ/n)$ where each edge is independently assigned a random signed weight. For two uniformly chosen vertices, we study the joint distribution of the total weights and hopcounts (number of edges) of the near-minimum weight paths connecting them. Under certain conditions on the weight distribution, which ensure in particular that these paths are typically of positive weight, we prove that the point process formed by the rescaled pairs of total weight and hopcount, converges weakly to a Poisson point process with a random intensity. This random intensity is characterized by the product of two independent copies of the Biggins martingale limit of certain branching random walk. This result generalizes the work of Daly, Schulte, and Shneer (arXiv:2308.12149) from non-negative to signed weights.

Minimum-Weight Path in a Sparse Erdős--Rényi Graph with Signed Weights

TL;DR

This work extends minimal-weight path analysis on sparse Erdős–Rényi graphs to signed edge weights, under a tilt condition that ensures path weights grow with length. By combining renewal theory, local weak convergence to branching random walks, and Chen–Stein Poisson approximation, the authors prove that the rescaled extremal pairs (weight, hopcount) converge to a Cox process with random intensity given by the product of two independent Biggins BRW martingale limits. The results generalize previous nonnegative-weight findings, handling sign-induced complexities via a localized BRW framework and a careful conditioning scheme. The approach yields a precise extremal process description and accommodates lattice vs non-lattice weight distributions, providing a robust method for first-passage percolation on sparse random graphs with general weights.

Abstract

We consider a sparse Erdős--Rényi graph where each edge is independently assigned a random signed weight. For two uniformly chosen vertices, we study the joint distribution of the total weights and hopcounts (number of edges) of the near-minimum weight paths connecting them. Under certain conditions on the weight distribution, which ensure in particular that these paths are typically of positive weight, we prove that the point process formed by the rescaled pairs of total weight and hopcount, converges weakly to a Poisson point process with a random intensity. This random intensity is characterized by the product of two independent copies of the Biggins martingale limit of certain branching random walk. This result generalizes the work of Daly, Schulte, and Shneer (arXiv:2308.12149) from non-negative to signed weights.

Paper Structure

This paper contains 13 sections, 14 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations and related work
  3. Roadmap of the proof
  4. Asymptotic of Expectations
  5. Preliminary: Renewal function.
  6. First moment asymptotic
  7. Local weak convergence to branching random walk
  8. Conditioned first moment asymptotic
  9. Limiting Extremal Process
  10. Good events
  11. Convergence of extremal process
  12. Vanishing of correlation terms
  13. Proof of Lemma \ref{['Stein_method']}

Key Result

Theorem 1.1

Under assumptions def-alpha and eq-assumption-4, the process $\mathcal{E}_n$ converges in law to $\mathcal{E}_\infty$ with respect to the vague topology on the space of measures on $\mathbb{R}^{-\infty}\times \mathbb{R}^{-\infty,+\infty}$. Moreover, $\mathbf{P}(\mathcal{E}_n = \emptyset) \to \mathbf

Figures (2)

  • Figure 1: Illustration of a free path $p=[1,u] \oplus (v_{i})_{i=1}^{\ell-1} \oplus [v,n] \in \mathcal{P}_{u,v}(\ell)$.
  • Figure 2: Illustration of $s_{i}'(p'|p)$ and $e'_{i}(p'|p)$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5: Chen--Stein method for Poisson Approximation
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem-renewal-function']}
  • Remark 2.2
  • ...and 18 more